You didn't mention it, but what's underneath the current flooring? Or is it in such bad shape that it's not economically feasible to repair/refinish?

LVP is good. But some of it is better than others as far as holding up to gouging, heels, heavy items dragged across it, etc. Do some shopping.

If you do go whole-floor with vinyl, make sure you have plenty of extra because it's possible that a tenant or even Ol' Sol (UV) may mar the finish enough that replacement planks will be really noticeable. Or know that you may be replacing it sooner in that event.

Another option is commercial modular carpet (carpet tile). Most of the modular sold in big-box home stores is not very good. But many places sell remainders of commercial jobs, discontinueds, odd lots, etc., which might be just fine for what you'd carpet out of 850 square feet of living area, kitchen, bath, etc.). The stuff holds up very well and if it's ever damaged beyond being able to be rinsed out/shampooed tile(s) it can be replaced.

[ETA: In my rental I installed 98cent/sq ft. laminate flooring just because I needed something cheap and fast. It has held up over years of people and dogs and wheelchairs. It's showing some wear now so I'd replace it. I suppose that could be another option for you outside of kitchen/bath/entry areas. Again, some laminate is better than others. I may just have been lucky that no leaky dog or clumsy tenant sped up the floor-replacement project for me.]

The seller claims there is hardwood under the current floor, but I would rather keep that to refinish when we sell the place in several years, as it can only be refinished a few times, and I'm not sure it is everywhere, currently the flooring is worn sheet vinyl and worn carpet. The house was built in the 40s so I won't be surprised if there are some asbestos tiles between the hardwood and what is on there now. I suppose no solution is perfect. We looked today at LVP and laminate that was advertised as "waterproof" and both seem like good options.

If there are hardwood floors in the home, then vinyl becomes an issue. Vinyl and wood do not like each other. Vinyl has been known to stain hardwood (especially hardwood that still has a finish on it). Vinyl is not normally low cost either.

Laminate flooring is your best bet - but it doesn't like kitchens. That being said, you know it is not permanent and only need a few years out of it (3-5??? years would be an honest service level in kitchens).

A water proof laminate is the best of both worlds but the price tag can be hefty. You will need to decide between budget, practicality and longevity. Assess HOW LONG you need the floor to last and then work backwards towards a budget goal.

@Cancork Floor, what are some examples of waterproof laminates?

I'd love some suggestions too, waterproof laminate seems like the ideal option if we can keep the cost under control. I'd rather spend a little more for something that will look and wear better, flooring is the most prominent feature in the renovation.

I know some landlords that swear by Allure vinyl planks.